Math 2.0

statement theorem

Lebesgue's criterion for Riemann integrability

Statement

Let a<ba < b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be bounded. Then ff is Riemann integrable if and only if the set of points where ff is not continuous has Lebesgue measure zero.

Remarks

A set E⊆RE \subseteq \mathbb{R} has measure zero if for each ε>0\varepsilon > 0 it is covered by countably many open intervals of total length less than ε\varepsilon. Countable sets have measure zero. So Thomae's function is integrable, and so is each monotone function. The Dirichlet function is discontinuous everywhere, so it is not integrable. Giuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.

Explanations

L1 human

A bounded function is Riemann integrable exactly when it is continuous at almost all points. The bad points must form a set of total length zero.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionLebesgue's criterion for Riemann integrability, statement, no evidence levelLebesgue's criterion …statement
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  1. Limit point of a set definition
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  7. Riemann integrable function definition
  8. Lebesgue's criterion for Riemann integrability statement

Sources

  1. Web https://en.wikipedia.org/wiki/Riemann_integral, Integrability

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